A131334 Triangle read by rows: A000012(signed) * A065941 as infinite lower triangular matrices.

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 3, 2, 4, 2, 1, 0, 1, 3, 3, 6, 4, 3, 1, 1, 0, 4, 3, 9, 6, 7, 3, 1, 0, 1, 4, 4, 12, 9, 13, 7, 4, 1, 1, 0, 5, 4, 16, 12, 22, 13, 11, 4, 1, 0, 1, 5, 5, 20, 16, 34, 22, 24, 11, 5, 1, 1, 0, 6, 5, 25, 20, 50, 34, 46, 24, 16, 5, 1

Offset

1,13

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Formula

From Andrew Howroyd, Sep 22 2025: (Start)

T(n,k) = Sum_{i=k..n} (-1)^(n-i)*A065941(i-1,k-1).

G.f.: x*y*(1 + y*x)/((1 + x)*(1 - x - y^2*x^2)). (End)

Example

First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 1;
  0, 1, 1, 1;
  1, 0, 2, 1, 1;
  0, 1, 2, 2, 2, 1;
  1, 0, 3, 2, 4, 2, 1;
  0, 1, 3, 3, 6, 4, 3, 1;
  1, 0, 4, 3, 9, 6, 7, 3, 1;
  ...

Prog

(PARI)
U(n, k)=binomial(n-(k+1)\2, k\2) \\ A065941
T(n, k)=sum(i=k, n, (-1)^(n-i)*U(i-1, k-1)) \\ Andrew Howroyd, Sep 22 2025
(PARI)
T(n)=[Vecrev(p) | p<-Vec((1 + y*x)/((1 + x)*(1 - x - y^2*x^2)) + O(x^n))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Sep 22 2025

Crossrefs

Row sums are A000045.

Cf. A065941, A131332, A131333, A131335, A131336.

Sequence in context: A214341 A381104 A281871 * A004602 A247418 A229899

Adjacent sequences: A131331 A131332 A131333 * A131335 A131336 A131337

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 29 2007

Extensions

a(56) onwards from Andrew Howroyd, Sep 22 2025

Status

approved